Towards non-perturbative BV-theory via derived differential geometry

TL;DR

This paper develops a comprehensive geometric framework using derived differential geometry to model non-perturbative BV-theories, integrating higher, infinite-dimensional, and derived structures for advanced field theories.

Contribution

It introduces a concrete model of derived differential geometry with formal derived smooth stacks, enabling the study of complex geometric spaces relevant to non-perturbative BV-theories.

Findings

Constructed a formalism combining differential cohesion and homotopical algebraic geometry.

Applied the framework to scalar field theory and Yang-Mills theory.

Provided new tools for non-perturbative analysis in quantum field theory.

Abstract

We propose a global geometric framework which allows one to encode a natural non-perturbative generalisation of usual Batalin-Vilkovisky (BV-)theory. Namely, we construct a concrete model of derived differential geometry, whose geometric objects are formal derived smooth stacks, i.e. stacks on formal derived smooth manifolds, together with a notion of differential geometry on them. This provides a working language to study generalised geometric spaces that are smooth, infinite-dimensional, higher and derived at the same time. Such a formalism is obtained by combining Schreiber's differential cohesion with the machinery of T\"oen-Vezzosi's homotopical algebraic geometry applied to the theory of derived manifolds of Spivak and Carchedi-Steffens. We investigate two classes of examples of non-perturbative classical BV-theories in the context of derived differential cohesion: scalar field theory and Yang-Mills theory.